A&A 505, 845–852 (2009) Astronomy DOI: 10.1051/0004-6361/200911886 & c ESO 2009 Astrophysics Orbital evolution of the μ and ν dust ring particles of Uranus R. Sfair and S. M. Giuliatti Winter UNESP-São Paulo State University, C.P. 205, Guaratinguetá, CEP 12516-410, SP, Brazil e-mail: [email protected] Received 19 February 2009 / Accepted 24 June 2009 ABSTRACT The μ and ν rings of Uranus form a secondary ring-moon system with the satellites Puck, Mab, Portia, and Rosalind. These rings are tenuous and dominated by micrometric particles, which can be strongly disturbed by dissipative forces, such as the solar radiation pressure. In the region of these rings, the solar radiation force and the planetary oblateness change the orbital evolution of these dust particles significantly. In this work, we performed a numerical analysis of the orbital evolution of a sample of particles with radii of 1, 3, 5, and 10 μm under the influence of these perturbations, combined with the gravitational interaction with the close satellites. As expected, the Poynting-Robertson component of the solar radiation force causes the collapse of the orbits on a timescale between 3.1 × 105 and 3.6 × 106 years, while the radiation pressure causes an increase in the eccentricity of the particles. The inclusion of Uranus’s oblateness prevents a large variation in the eccentricity, confining the particles in the region of the rings. The encounters with the close satellites produce variations in the semimajor axis of the particles, leading them to move inward and outward within the ring region. These particles can either remain within the region of the rings or collide with a neighbouring satellite. The number of collisions depends on the size of both the particles and the satellites, and the radial width of the ring. For the time span analysed, the percentage of particles that collide with a satellite varies from 43% to 94% for the ν ring, and from 12% to 62% for the μ ring. Our study shows that all collisions with Portia and Rosalind have the value of impact velocity comparable to the escape velocity, which could result in the deposition of material onto the surface of the satellite. Collisions between Puck and particles larger than 1 μm also occur at an impact velocity comparable to the value of the escape velocity. The exception is Mab, which is hit by particles with velocities several times larger than the escape velocity. These collisions are energetic enough to eject material and supply material to the μ ring. However, only a few particles (3%) collide with the surface of the satellite Mab at such a velocity. Key words. planets: rings – planets and satellites: individual: Uranus – methods: N-body simulations 1. Introduction Until 1977, the only known planet surrounded by rings was Saturn. In this year, during the occultation of the star SAO 158687, four narrow rings around Uranus were discovered (Elliot et al. 1977), and further observations from Voyager II found six more rings and faint dust bands between the rings (Smith et al. 1986). The ring orbits at 51 149 km (2 Rp), the outer edge of the main ring system. Showalter & Lissauer (2006) obtained several images with long exposure times of the Uranian system using the Hubble Space Telescope (HST), which discovered two rings, μ and ν, which represented a secondary ring-moon system orbiting out- side the main rings. Both rings are very faint and their radial pro- files, defined in terms of the brightness of the ring as a function of the distance from the planet’s centre, show a distinguished Fig. 1. Schematic view of the secondary ring-moon system of Uranus. triangular profile. Their orbits are closely related with satellites: The solid lines correspond to the satellites orbits and the rings are rep- the peak, which corresponds to the distance from the planet’s resented by shadow areas. The planet (shadow area until 1 Rp)andthe centre of maximum normal value of I/F (measurement of the ring are also indicated. All distances are scaled in units of Uranus’ brightness of the ring that would be observed from a perpendic- radius (Rp). ular point of view), of μ ring is almost aligned with the orbit of the satellite Mab, while the inner edge of the ring coincides with 2 Rp indicates the position of the ring, the outermost ring of the the orbit of Puck. The ν ring peak does not match the orbit of main system. any satellite, but the ring orbits between the satellites Rosalind The μ and ν rings are radially wide and predominantly for- and Portia. Figure 1 shows a schematic view of the μ and ν rings ward scattering, indicating that they are dominated by micro- and the closest satellites. The radial extension of each ring is metric dust (de Pater et al. 2006). Dust particles can be disturbed represented by shadow areas, and the solid lines correspond to by several non-gravitational forces, such as the solar radiation the orbits of Puck, Mab, Rosalind, and Portia. All distances are force, the solar tidal, and the planetary oblateness. These forces represented in units of Uranus’ radius (Rp), and the solid line at are much weaker than the gravity force of the planet. However, Article published by EDP Sciences 846 R. Sfair & S. M. Giuliatti Winter: Orbital evolution of the μ and ν dust ring particles of Uranus they change the orbital energy of the particle, which can be im- portant to its long-term evolution. In this paper, we analyse the orbital evolution of a sample of dust particles, located at the secondary ring system of Uranus, affected by the solar radiation pressure, the oblateness of the planet, and the gravitational perturbations of the satellites. In Sect. 2, we analyse the strength of the solar radiation pressure and the oblateness of the planet, and in Sect. 3, we describe the numerical method. The effects of these perturbations on the ring particles and the orbital evolution of them are analysed in Sects. 4 and 5, respectively. Section 6 deals with the collisions between the particles and the satellites. Our final conclusions are presented in the last section. 2. Force parameter Hamilton & Krivov (1996) presented a way to compare the strength of the perturbation forces using dimensionless param- Fig. 2. Parameters values as a function of the distance of the planet: A eters that depend on the dust particles’ semimajor axis, and the (solar tide), W (planetary oblateness), and C (radiation pressure). The physical properties of the dust and the planet. Following their vertical bars denote the radial extent of the μ and ν rings. We assumed a notation, we calculated the values of the solar tidal (A), oblate- spherical grain with radius of 1 μm and uniform density of 1.0gcm−3. ness (W), and radiation pressure (C) parameters. The solar tidal parameter A is defined as i ≤ γ,whereγ is the obliquity of the planet (Hamilton 1993). In 3 n A = s , (1) these cases, the problem can be treated as bidimensional, when 4 n the Sun and the ring particle are located in the same plane. where n is the mean motion of the planet around the Sun and n Although Uranus’s dusty rings describe equatorial orbits, s ◦ is the mean motion of the particle. the obliquity of Uranus is 97.8 (Murray & Dermott 1999). The parameter W, which describes the strength of the plane- Therefore, this problem requires a three-dimensional approach, tary oblateness, is written as since the Sun and the ring particle do not share the same orbital 2 plane. 3 R ns For a particle that describes an orbit around a planet with W = J2 , (2) 2 a n obliquity γ, the components of the solar radiation force, in a where J2 denotes the second zonal harmonic coefficient, R is the Cartesian inertial frame centred on the planet, can be written as radius of the planet, and a is the semimajor axis of the particle. (Mignard 1984) The ratio σ of the radiative force to the planets’ gravity of a particle with radius r and density ρ is ⎡ ⎤ β ⎢ 2 v v v v ⎥ GMs ⎢ xs xs x xs x ⎥ 3 Fsa F = ⎣⎢cos (n t) − + − + ⎦⎥ , (5) σ = Q . (3) x 2 s pr ρ rsp rsp c c c c 4 GMsc r ⎡ ⎢ 2 βGM ⎢ y vys vy In this equation, Qpr is the radiation pressure coefficient, Fs is s ⎢ s Fy = ⎣⎢cos(γ)sin(nst) − + the incoming solar flux at the planet, GM is the planet’s gravi- r2 r c c s sp sp tational constant, and c is the speed of light. vys vy Using the previous definition of σ, the perturbation caused − + , (6) c ⎡ c by radiation pressure is written as ⎢ 2 βGMs ⎢ zs vzs vz 3 n F = ⎣⎢sin(γ)sin(n t) − + = s σ. z 2 s C (4) rsp rsp c c 2 n v v All of these parameters were calculated for a dust particle − zs + z , (7) around Uranus, and the results are presented in Fig. 2. c c As can be seen in Fig. 2, the solar tide is relevant to particles where rsp is the module of the Sun-planet vector, and (xs, ys, zs) located faraway from the planet (>50 Rp), thus it can be safely are the components of rsp. The components of the velocity of the ignored in the region of the μ and ν rings. On the other hand, v v v v v v ff particle are ( x, y, z), and ( xs, ys, zs) are the components of for these rings the e ects of radiation pressure and oblateness the velocity of the planet around the Sun.